Working on supplemental section

	\begin{figure}[h]

		\centering

		\includegraphics[width=\textwidth]{GeneratedFigures/01_ADF_BF.pdf}

		\caption{\label{fig:simul_adf}\textbf{Collected Images}} 

		\caption{\label{fig:simul_adf}\textbf{Collected Images. a,} 

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		Summed 4D-STEM image obtained by summing the diffraction patterns from all the 4D-STEM frames, at every scan position. 

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		\textbf{b,} Simultaneous ADF-STEM image.}

	\end{figure}

	\begin{figure}[h]

		\centering

		\includegraphics[width=\textwidth]{GeneratedFigures/01_ABF_mrads.pdf}

		\caption{\label{fig:different_ABFs}\textbf{Multiple synthetic ABF-STEM images with different collection angles.}} 

	\end{figure}

	\begin{figure}[h]

		\centering

		\includegraphics[width=\textwidth]{GeneratedFigures/01_ADF_gradient.pdf}

		\caption{\label{fig:adf_gradient}\textbf{Particle identification from ADF images. a,} 

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		Original ADF-STEM image. 

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		\textbf{b,} Gradient of the ADF-STEM image. 

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		\textbf{c,} The particle boundaries calculated from the gradient image overlaid in red over the ADF-STEM image.}

	\end{figure}

	\begin{figure}[h]

		\centering

		\includegraphics[width=\textwidth]{GeneratedFigures/01_particle_positions.pdf}

		\caption{\label{fig:particle_ROIs}\textbf{Particle ROIs. a - e,} 

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		Individual regions of interest of the five different particles overlaid on the ADF-STEM image.}

	\end{figure}

	\section{Mapping volumetric strain in a single off-axis nanoparticle}

	\section{Generating combined diffraction information}

	\section{Correcting projector lens distortions}

		\begin{figure}[h]

			\centering

			\includegraphics[width=\textwidth]{GeneratedFigures/linear_corr.pdf}

			\caption{\label{fig:linear_corr}\textbf{Linear Correction. a,} Subset of points linearly fitted, shown in cartesian co-ordinates. \textbf{b,} Points shown in \autoref{fig:linear_corr}\blu{a} shown in polar co-ordinates, where the waviness arising due to aberrations are clearly visible.} 

			\caption{\label{fig:linear_corr}\textbf{Linear Correction. a,} 

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			Subset of points linearly fitted, shown in cartesian co-ordinates. 

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			\textbf{b,} Points shown in \autoref{fig:linear_corr}\blu{a} shown in polar co-ordinates, where the waviness arising due to aberrations are clearly visible.} 

		\end{figure}

		However, there is a problem - cleanly visible in \autoref{fig:all_points}\blu{b}. There are fundamentally five peaks - and the radial distances of those peaks should be pretty constant with respect to the angle. However, there is a waviness in the data, which broadens the peaks and makes the data very difficult to interpret. Like, look at the peak at $\mathrm{\approx 70}$ pixels in \autoref{fig:all_points}\blu{c}. It is split into two - because of the waviness. These distortions have been observed before too, and their influence on the data analysis noted by Mahr \textit{et al.} \cite{4dstem_distortions}.

		However, there is a problem - cleanly visible in \autoref{fig:all_points}\blu{b}. 

		%

		There are fundamentally five peaks - and the radial distances of those peaks should be pretty constant with respect to the angle. 

		%

		However, there is a waviness in the data, which broadens the peaks and makes the data very difficult to interpret. Like, look at the peak at $\mathrm{\approx 70}$ pixels in \autoref{fig:all_points}\blu{c}. 

		%

		It is split into two - because of the waviness. 

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		These distortions have been observed before too, and their influence on the data analysis noted by Mahr \textit{et al.} \cite{4dstem_distortions}.

		And the source of this waviness is post-specimen lens aberrations, which we have to correct. One way to express the aberrations is through \autoref{eq:ps_aberrations}.

		And the source of this waviness is post-specimen lens aberrations, which we have to correct. 

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		One way to express the aberrations is through \autoref{eq:ps_aberrations}.

		\begin{equation}\label{eq:ps_aberrations}

			r_i^a = r_i^0 \times \left[\Pi_{i=1}^{n}\left(1 + a_i\sin\left(\frac{\theta_i}{n}\right)\right)\right]

		\end{equation}

		where $r_i^0$ is the real radial distance, $r_i^a$ is the measured radial distance due to aberrations, and $n$ is the aberration order. Thus, one way to think of this is that there are multiple sine waves. The frequency of the sine wave is the aberration order. Also, all the sine waves are not co-located, which is an issue. So, we need to build an aberration corrector function that does this. Secondly, why is this multiplicative? If you compare the peaks at 70 pixels versus those at 100 pixels in \autoref{fig:all_points}\blu{b}, the higher radius peaks are wavier. Thus, the effect is radially dependent, and not a constant, which means multiplicative.

		where $r_i^0$ is the real radial distance, $r_i^a$ is the measured radial distance due to aberrations, and $n$ is the aberration order. 

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		Thus, one way to think of this is that there are multiple sine waves. 

		%

		The frequency of the sine wave is the aberration order. 

		%

		Also, all the sine waves are not co-located, which is an issue. 

		%

		So, we need to build an aberration corrector function that does this. 

		%

		Secondly, why is this multiplicative? 

		%

		If you compare the peaks at 70 pixels versus those at 100 pixels in \autoref{fig:all_points}\blu{b}, the higher radius peaks are wavier. 

		%

		Thus, the effect is radially dependent, and not a constant, which means multiplicative.

		To test the basic nature of aberrations, we chose to do a linear correction based on the modified \autoref{eq:lin_corr}.

		\begin{equation}\label{eq:lin_corr}

			r_i^a = r_i^0 + \Sigma_{i=1}^{n}\left(a_i\sin\left(\frac{\theta_i}{n}\right)\right)

		\end{equation}

		The results from following such a correction is demonstrated in \autoref{fig:linear_corr}\blu{a} and \autoref{fig:linear_corr}\blu{b}, where the linear correction was performed only on a subset of the data. This subset belonged to BOL particles, and positions away from particle shell where maximum strain variation is expected. Also, only those points that were between 80 to 100 pixels from the center were chosen, thus ensuring only one peak value was represented.

		The results from following such a correction is demonstrated in \autoref{fig:linear_corr}\blu{a} and \autoref{fig:linear_corr}\blu{b}, where the linear correction was performed only on a subset of the data. 

		%

		This subset belonged to BOL particles, and positions away from particle shell where maximum strain variation is expected. 

		%

		Also, only those points that were between 80 to 100 pixels from the center were chosen, thus ensuring only one peak value was represented.

		While this demonstrated that we are correct on the technical nature of the aberrations, this does not correct the whole dataset. So, what we need is to do this with non-linear (multiplicative) correction on the entire dataset. To do this, we need to find the parameters $a_i$ and $r_i$ that best fit the data. This is done using the Expectation-Maximization (EM) algorithm\cite{em_algo}. The EM algorithm is a two-step process. In the first step, we calculate the expectation of the data, given the current parameters. In the second step, we calculate the maximum likelihood of the parameters, given the current expectation. This is repeated until the parameters converge. The expectation is calculated using the following equation.

		While this demonstrated that we are correct on the technical nature of the aberrations, this does not correct the whole dataset. 

		%

		So, what we need is to do this with non-linear (multiplicative) correction on the entire dataset. 

		%

		To do this, we need to find the parameters $a_i$ and $r_i$ that best fit the data. 

		%

		This is done using the Expectation-Maximization (EM) algorithm\cite{em_algo}. 

		%

		The EM algorithm is a two-step process. 

		%

		In the first step, we calculate the expectation of the data, given the current parameters. 

		%

		In the second step, we calculate the maximum likelihood of the parameters, given the current expectation. 

		%

		This is repeated until the parameters converge. 

		%

		The expectation is calculated using the following equation.

		\begin{figure}[h]

			\centering

			\includegraphics[width=\textwidth]{GeneratedFigures/corr_rings.pdf}